Saturday, January 5, 2013

Simmechanics: Simulating floor interaction/collision


I'll demonstrate how to simulate a floor in Simmechanics, this floor will prevent free falling (6 DOF) objects from falling below a threshold level ("the floor level"), i.e. I will simulate a collision/interaction between the free falling object and the floor.

I'll assume you have a basic understanding of how Simmechanics works. If you don't, you might want to check this post.

The problem


If you have used Simmechanics before, then you know it's an excellent toolbox for simulation of multibody dynamics. You can easily work with robotic arms, which are always fixed to the ground. But what happens when you want to work with legged robots, that can't be fixed to the ground. Well, then you run in a problem, if you don't fix any part of the robot to the ground, then the robot will fall and fall by action of the gravity. Removing the gravity is not an option, because you want to design the controllers of your robot with gravity compensation, even worse your robot will float if you simulate a zero gravity environment.

Let's develop a minimal model to illustrate the problem: a 6 DOF ball.

Illustrating the problem


We'll start by drawing the floor. Implement the following block diagram in Simulink.

Block diagram to visualize the floor.

Modify the Floor block parameters to match the following image, and also on the Visualization tab change the Body geometry option to "Equivalent ellipsoid from mass properties".

Parameters to draw a 1 meter radius floor.

Next, start a simulation. Activate the the Isometric View and the "Enable Automatic Expanding Fit".

Animation showing the floor as a 1 m radius disk.

Now, we add the free falling ball. Add more blocks to the model to match the following diagram.

Added a free falling ball to the model.

Next, modify the parameters of the Above Ground block and the Ball block, to match the following images. On the Visualization tab of the ball block also select "Equivalent ellipsoid from mass properties" on the Body geometry option.

Above Ground parameters
Ball parameters

Now simulate the system and observe the problem.

Ball falling through the floor.

The Idea


Known the problem, we must now create a solution. Let's begin by analyzing how the floor behaves. Ideally the floor is a rigid body that exerts enough force over adjacent bodies to prevent them from falling through. Now, perfect rigid bodies doesn't exist in the real world, all bodies deform when stressed or compressed. This means our floor model must be elastic, this can be modeled as a spring.

Our floor model can't be modeled only as a spring, otherwise will end with a trampoline. When we fall to the ground, we don't jump back to the sky, instead our potential energy dissipates as heat or as permanent deformation. This means our floor model must also have a dissipation element.

A mass-spring-damper system.

That description resembles a lot to a mass-spring-damper system. As you might know those system tend to vibrate, unless they are overdamped. This means our floor model must behave as an overdamped mass-spring-damper system. The final condition is that the free falling object must be the mass of the system, this means that our floor model doesn't really has a mass parameter. Other way to think of it is that the mass is detachable, and this mass is only affected by the spring and the damper when is at or below the floor level.

Implementing the floor interaction


First, let's enable some sensor/actuator ports on the free falling ball. These two ports must be located at the same point, at the bottom of the ball.

Adding sensor/actuator ports to the free falling ball.

The floor control diagram can be implemented as follows:

Floor Control Diagram.

The body sensor is configured to output the position in meters, while the body actuator is configured to exert force in newtons over the ball. Using a demuxer, we extract the Y component of the position, then we compare it to the floor level. If the Y position of the bottom of the ball is below the floor level, then we activate the spring-damper system represented by the gains 'k' and 'c', otherwise the floor exerts no force over the ball. The force developed by the spring damper system is exerted over the ball but only in the Y direction, thanks to a muxer and the body actuator.

The final step is selecting 'k' and 'c'. You need to considerate the following things:

  • The damping ratio "c / 2 / sqrt(m * k)" should be greater than 1, to guarantee overdamping.
  • The free falling body will sink a distance "m * g / k" into the ground, so you might want to increase 'k', right?
  • But Increasing 'k', also increases the computation load of the simulation.

How do I select the parameters 'k' and 'c'? I first select 'k' balancing how much sinking and computation load can I tolerate. Then I compute 'c' using the biggest mass and choosing a damping ratio of 1.

The Results


I'll leave you with the simulation output.

Floor control enabled.

Ball interacting with the floor. (Y position)

As a closing remark, I must say that I have used this method in the simulation of a humanoid robot with positive results.

144 comments:

  1. Jorge,

    How did you go about modelling the floor? What values did you input for the blocks in your SimMechanics block diagram?

    ReplyDelete
    Replies
    1. Hey there,

      For the spring constant 'k', I used the floor material (say concrete) young's modulus times the "contact area", and for the friction coefficient 'c' I selected a value that results in critically damped oscillation.

      Hope that helps

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  2. Hey,
    I just stumbled over your site and find your Simulations very interesting!
    I am currently trying to get into Simmechanics but have a few starting problems.
    Can you please send me the mdl file of this simulation?
    that would really be of great help for me!

    Kind regards, Thomas

    ReplyDelete
  3. Hey Jorge,

    thanks so much for your nice explanation. I am writing my diploma thesis about pretty much the same thing as you did with your humanoid robot simulation and this tutorial just blew away one of the many problems i am encountering. ;)

    In addition i think i found a little mistake in the explanation above:

    The equation for calculating the damping ratio you mentioned above "c / 2 / sqrt(m * k)"
    should look like this "c / (2*sqrt(m * k))"

    Cheers

    ReplyDelete
  4. great work. can u share the value of k and c in your system ?
    thank you so much

    ReplyDelete
  5. I build up this model in simmechanics model. But unfortunately with the "derivative" block (the forceconsisted from spring and damping was almost proportional and ball fell down) it didn't work properly so i exchange it to the "Switched
    derivative for
    linearization" block and it started to work. Any idea why?

    ReplyDelete
  6. Hola Jorge, Estoy Siguiendo tu Blog, y Este trabajo de simular un piso en SimMechanics es muy bueno, favor me pordías ayudar de como Simular piernas de Robot Bípedo para la Camita, Agradeceria Mucho tu ayuda.

    Saludo,
    Josmell Alva.

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    เมื่อคุณอยากเข้ามาเล่นเกมคาสิโนออนไลน์ sa บาคาร่า สามารถเข้ามาเรื่องเล่นเกม sa king789 ซึ่งวันนี้พวกเราจะมาบอกถึงความต่างสำหรับการเล่นเกมบาคาร่ากับ sa king789
    • sa king789 เป็นเว็บไซต์เล่นเกมบาคาร่าออนไลน์เชื่อใจได้ ด้วยเหตุว่าเป็นเว็บไซต์ที่เปิดให้บริการมานานมาสมาชิกกว่าหมื่นคนที่ใครๆก็รู้จักนั้นเชื่อมั่นได้เลยว่าเล่นเกม sa บาคาร่ากับพวกเราได้เงินแน่ๆ
    • คุณจะได้รับโปรโมชั่นออนไลน์ที่น่าดึงดูดมากมายก่ายกองเมื่อเข้ามาเล่นเกมบาคาร่าที่นี้ เพื่อเป็นการเอาอกเอาใจเหล่าสมาชิกเว็บไซต์ พวกเรามีโปรโมชั่นแจกเครดิตฟรีที่จะให้ท่านได้ประหยัดเงินสำหรับที่คุณจะเอาไปลงทุนเล่นเกมบาคาร่า ทำให้ท่านทำเงินได้มากขึ้นกว่าเดิม
    • ระบบเกมของ sa king789 เชื่อถือได้ทุกเกมอ เพราะเหตุว่าเป็นเกมที่ผลิตขึ้นมาจากรชนักนักเขียนโปรแกรมชั้นแนวหน้า
    • เว้นแต่บาคาร่าแล้ว sa king789 ยังมีเกมคาสิโนอื่นๆที่น่าดึงดูดมากมายให้ท่านได้ทดลองเล่น ไม่ว่าจะเป็นเกมไพ่ บาคาร่า สล็อต รูเล็ต ก็มีให้ท่านได้เลือกเล่นตามอยาก
    • sa king789 มีบริการฝาก ถอน อัตโนมัติ ทำให้ผู้เล่นเกมได้รับความสบาย สบายเยอะขึ้นเรื่อยๆไม่ว่าจะฝาก หรืออยากได้เบิกเงินที่ได้จากการเล่นเกมบาคาร่า
    น่าเชื่อเลยว่าถ้าเกิดคุณอยาก หรือพอใจที่จะเข้ามาเล่นเกมบาคาร่ากับ sa king789 คุณจะได้เจอกับความสนุกสนาน ความระทึกใจ และก็สิทธิพิเศษดีๆอีกเพียบเลยที่สามารถช่วยให้ท่านบันเทิงใจ และก็สร้างรายได้จากเว็บไซต์คาสิโนออนไลน์ของพวกเรา แต่คุณสามารถเข้ามาถามไถ่ข้อมูลเพิ่มเติมเกี่ยวกับการเล่นเกมบาคาร่าออนไลน์ถึงที่เหมาะเว็บไซต์คาสิโนออนไลน์ของพวกเรา พวกเรามีคณะทำงานที่จะรอให้การช่วยเหลือในทุกกรรมวิธีสมัครเล่นเกมให้กับคุณ รวมทั้งยังมีแนวทางการเล่นเกม sa ให้ลุกรได้เข้าไปเรียนเพื่อเพิ่มช่องทางทำเงินให้กับคุณได้ ถ้าเกิดคุณไม่ได้อยากพลาดความพิเศษนี้รีบมาสมัครสมาชิกกับ sa king789 ได้เลย
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