Modeling Workflow

In the introduction, we looked at the structure of an omni-directional wheel and how it works. We examined how this wheel design allows the robot assembly to move in any direction. It does this without changing its orientation. It also achieves holonomic motion. Now, it’s time to translate that information into an actual virtual model. This model acts and behaves exactly like the real thing. This is where Simscape comes into the picture.

We will use the Multibody modeling tools in Simscape to build this model. We will hook up this model to control algorithms developed in Simulink and MATLAB. This model can also be used as a digital twin for the actual robot that I am building for my lab. I will talk about digital twins at length in another post.

In this post, we are going to build our robot model in a step-by-step fashion, as follows:

1. Breakdown the model into a hierarchical structure
2. Arrive at a flow-chart for building the parts
3. Build steps:
   • Passive wheel and axle
   • Wheel hub assembly
   • Finish Wheel assembly

We will attach the wheels to the robot chassis and model the interaction with the ground plane in the next blog entry.

If you would like to first go take a look at the basics of Multibody modeling in Simscape, follow the link to the documentation. You can find it in the getting started section at the bottom of the page..

Breakdown the model into a hierarchical structure

Let us start by breaking down the robot assembly into a hierarchical structure. This is how that structure looks conceptually.

Arrive at a flow-chart for building the parts

Starting out on a modeling project feels daunting when you look at the assembly as a whole. Still, once you have broken it down into smaller parts, it’s easy to see the repetitive elements. You need to model them only once. Also, as we model the smaller parts, the steps become more intuitive. When we assemble them into the next higher order part, the process becomes more straightforward. This is how the flowchart is actually going to look like once we get started.

Modeling the Passive wheel with axle

Before we start modeling the wheel, we are going to need some 3D models to represent its components. These components are the passive wheel and the wheel hub. Here is the STL file for the passive wheel used in this project:

Now that we have the 3D model for the wheel, let’s create a File Solid block in the Simscape model. First, set the unit type to Custom. Then, select the downloaded STL file. Set the unit type to Custom and select the downloaded STL file. You should see the wheel rendered in the preview window. Name the block as Wheel.

Make sure to check the “Convex Hull” option in the export section under Geometry. This is important. This allows the wheel to interact with other surfaces by providing a collision model. This collision model can be used by the physics engine.

You will notice that the wheel has a hole along it’s longer axis. We will align a cylindrical axle with this hole and use it to model the freely-rotating wheel motion. To achieve this, we will use a revolute joint and call it AxleToWheel Joint. We will use the default settings for this joint. Its rotation axis is already aligned with the Z-axis of the wheel. The reference frame for this setup is connected to an output port called R. The convex hull frame from the wheel is connected to an output port called CH.

Now, we can convert this assembly into a subsystem so that we can reuse it in our larger model.

Modeling the Wheel sub-assembly

Now that we have a passive wheel part, we can use it to build our next part. This is the wheel sub-assembly consisting of a wheel hub and six passive wheels distributed along its circumference. The 3D part is shown below.

To start, let’s go ahead and add a File Solid block. Then, load the wheel hub STL file we downloaded earlier. Finally, call it Wheel_hub.

Let us start with one passive wheel and connect it to the hub with an appropriate transform. Once we have fine-tuned this connection, we can replicate it for the rest of the passive wheels. This is how the connection looks like:

The two transform blocks orient the passive wheel and hub with respect to each other and the reference frame. Here are the properties of each of the transform blocks:

The transform from the reference frame to the hub rotates the hub along the +Z-axis. This is done to align one of the ports to align horizontally. We will position the passive wheel to fit inside this port.

The transform from the reference frame to the passive wheel translates the wheel to its actual position. This translation ensures the passive wheel fits exactly inside the horizontally aligned port on the hub. This is how the assembly should look at this point:

The transform also aligns the +Z-axis for the passive wheel to the +X-axis of the hub. Similarly it aligns the +Y-axis of the passive wheel to the -Y-axis of the hub. This ensures that the passive wheel rotates freely inside the port it is aligned with in the correct direction.

Now, we can test the motion of this passive wheel and confirm its rotation. To do that, we must temporarily set the actuation property on the AxleToWheel_Joint to accept torque as an input. This adds an extra input port to the joint block called T.

Now, we can apply a constant torque to this joint and run the simulation. If everything is setup correct, we should see the passive wheel spin inside the hub port. To apply a constant value to this torque input port, we need to temporarily add a PS Constant block. This is how the connection should look like:

Now, we can run the simulation and we should see this:

Phew! Now that is some serious modeling work we just accomplished here. It is easy to see that the wheel on the opposite side has the same orientation. It is translated in the opposite direction. So, we can duplicate the blocks for the first wheel and just change its RefToWheelTransform block. We can now treat these two wheels as a pair. It is easy to see that the other two pairs are just rotated versions of this pair. By duplicating the pair and rotating it by 60 and 120 degrees, we can create the remaining wheels. By duplicating the wheels from the original one, we get the same axis alignment on all of them. This is how the final setup should look like:

Now, we have the wheel sub-assembly ready to go. This is how it should look like at this point:

Now, that we have the finished wheel sub-assembly, lets convert it into a reusable subsystem.

Finishing the wheel

In the introduction, we spoke about why we need two wheel sub-assemblies joined together. Now, lets get down to building the whole wheel with the sub-assembly we already created.

To finish the assembly, we start with joining two copies of the wheel sub-assembly with a Fixed or Weld joint. We apply a rotation transform to one of the sub-assemblies to rotate it by 30 degrees. The wheel is now finished structurally. We still need to add a way to rotate these wheels by applying external torque. We do this by adding a revolute joint and connecting it to the joined assembly. We enable the torque input in the actuation section and export the angular velocity in the sensing section. This allows us to connect a motor torque to the wheel, and sense its rotation.

And with this, the entire omni-directional wheel is now ready. We have not spoken about the blocks with dotted lines in the model at all. This is by design. We will talk about the convex hull and geometry connections when we get to the step where we want to simulate the motion of the entire robot on a ground plane. But, for now, sit back and sip on some coffee. You have earned it.

You can find the library model containing the finished wheel models in the GitHub repository link below.

https://github.com/artineering/omni-directional-robot

Getting Started with Simscape

There is very good documentation with examples to learn the basics of Multibody modeling with Simscape provided here: https://www.mathworks.com/help/sm/index.html?s_tid=CRUX_lftnav

3D Models

You can download the models from here: https://cults3d.com/en/3d-model/gadget/omni-wheels-for-robotics.

Introduction

Every robot navigates with different levels of control and agility. If a mobile robot’s level of control matches its total possible degrees of freedom, its movement is described as holonomic[1]. When the level of control is less than its available degrees of freedom, its movement is described as non-holonomic. A good example of this is a car. The car has three available degrees of freedom – its position in the XY plane and its orientation. Still, it can only control its movement with two controls – acceleration and steering angle. It’s easy to imagine the limitation this imposes on the available paths a car can follow.

A standard wheel only has one degree of freedom, i.e. it can only rotate about its center. This rotation causes the wheel to exert a force along its tangent on the road. The road exerts a reactionary force due to friction in the opposite direction. The weight of the robot exerts a force downwards on the road. When the force exerted by the wheel is lesser than the frictional force, the wheel rolls. If the force exerted by the wheel exceeds the frictional force, the wheel will lose contact with the road. It will then rotate freely, what we also call as skidding. Next time you are thinking about creating some figure 8s in an empty parking lot, think about this. How much does this force need to exceed the frictional force to make the tires squeal?

We can add another degree of freedom to this standard wheel. This can be achieved by adding a small wheel between the wheel and the road. This setup allows the wheel to roll in a direction perpendicular to the rolling direction of the larger wheel. Now, the wheel can continue rolling along the primary direction of motion by applying a rotational torque. It can also roll in the secondary direction. This is done by applying a lateral force along the rotational axis of the larger wheel. Since we don’t apply any external force to this smaller wheel, it is called a passive wheel.

While this does add an extra degree of freedom, the rolling motion of the larger wheel is uneven. This is due to the cylindrical shape of the passive roller. Also, it is easy to see that one would have to add more such passive rollers along the circumference. This is necessary to achieve this extra degree of freedom at any given rotation of the larger wheel. Furthermore, we can change the shape of the passive rollers. Its profile can exactly match the circumference of the larger wheel. This gives it a barrel shape.

Adding the barrel-shaped rollers helps keep the overall shape of the wheel. Still, there are still gaps between the rollers. In these gaps, the wheel can’t roll freely. One solution explored to solve this problem is to make the rollers as small as possible. This makes the gap between them negligible. Another clever solution to this problem is to simply sandwich two of these assemblies together. Then, rotate them by a fixed angle compared to each other. The rendering shows a wheel configuration with six passive rollers. Choice of a particular configuration will largely depend on cost and application area.

To finish off this section, we will finally try to answer this question. How does this wheel help in making the robot omni-directional? The wheel we arrived at has two degrees of freedom. Adding three such wheels to a robot chassis allows us to add the same two degrees of freedom. This applies to the entire robot as well. To understand how such an assembly would move, let us consider the following scenarios:

Let us start by discussing the easiest scenario (e). In this case, we apply an equal rotation to all three wheels in the same direction. This causes the entire robot assembly to rotate in place. In the rotation amounts are different, the robot will move in a spiral as seen in (f).

In scenarios (a) & (b), we apply opposite equal rotations to two wheels. This pushes or pulls against the third wheel which is unpowered. Due to the passive rollers, this wheel will freely roll in the direction that the assembly is pushing or pulling. In this case, it is along its own main axis of rotation.

In scenarios (c) & (d), different combinations of rotations to each wheel will result in lateral movement of the entire assembly. In fact, we can show that by carefully choosing the right rotation amount and direction for each wheel, we can move the entire assembly in any direction. This does not change its original orientation, making it an omni-directional robot.

Now we have the basic idea in place about holonomic drives and omni-directional robots. We have seen how the omni-directional wheels behave given different inputs. We have also seen how these wheels allow the robot to move in any direction. And we have done so without a single line of mathematics. But as we start to move to the exciting step of modeling the assembly and its behavior in Simscape, we will dive into some of that math. We can’t avoid it.

References

1. M. West and H. Asada, “Design of a holonomic omnidirectional vehicle,” Proceedings 1992 IEEE International Conference on Robotics and Automation, Nice, France, 1992, pp. 97-103 vol.1, doi: 10.1109/ROBOT.1992.220328. keywords: {Vehicles;Wheels;Angular velocity;Tires;Kinematics;Jacobian matrices;Friction;Manufacturing industries;Angular velocity control;Mechanical systems},