This Christmas holiday I started working on a new robot, called Agilis. This robot should be a very agile and elegantly moving robot. The frame is based on a triangle equipped with holonomic wheels. So you might think, “What’s new, it is like your last robot?”. From the outside this is true, but it gets new and better brains on the inside. Let me tell you what I envision.

Most robots I built went from point A to point B, only then to decide what to do next. Others just drove around avoiding obstacles. This one should be able to do both at the same time. Agilis must be able to perform complex manouvres, like driving in a straight line while rotating around its centre, or like driving an arc while keeping pointed at an arbitrary spot. It should constantly use sensory input to stay on course, or to alter its course if needed. And all this must go fluently, just like a cat walking through the room.

Over the next several posts I will discuss the different aspects of Agilis. This first post deals with the drive system.

## the chassis

Agilis is a three wheeled holonomic platform. This means it can drive in any direction without turning. It can turn around any point, even around its own center. Each wheel is driven by a NXT motor via a gear train that has a 1:2 ratio, the wheels rotate twice as fast as the motors. This makes Agilis a relatively fast robot. The gear train has to be very sturdy to handle the torque of the motors. It also has to be precise to allow for odometry. I used the same setup that I developed for my last robot, Koios the Guard bot.

## From robot speed to motor speed

It is not very easy to control a holonomic robot, it takes some math. I created a kinematic model that does just that. The model takes robot speed as input and gives motor speed as output. Robot speed is expressed as speed in x-direction, y-direction and rotational speed. Motor speed is expressed as encoder ticks per second.

So how does this kinematic model look like? For a single wheel this looks like a function that takes the three robot speeds as input. For the three wheels together it looks like a matrix multiplication that multiplies a robot speed vector {xSpeed,ySpeed,angularSpeed} with a kinematic matrix. The resulting vector containers the speed of each of the three wheels. Let’s take a look at the single wheel function first.

To translate robot speed into motor speed one needs to know some physical aspects of the robot, the wheel and the motor. How big is the wheel, how far is it from the center of the robot, under what angle is it mounted, what is the gear ratio of the gear train and what is the number of encoder ticks per full cycle of the motor? With all this information one can write a formula to calculate motor speed from robot speed. Here is the formula.

motorSpeed =
xSpeed * (cosine(wheelAngle) * nEncoderTicks / ( gearRatio * 2 * PI * wheelRadius) -
ySpeed * (sinus(wheelAngle) * nEncoderTicks / (gearRatio * 2 * PI * wheelRadius) +
angularSpeed * distanceToCenter * nEncoderTicks / (gearRatio * 2 * PI * wheelRadius)

This formula might look daunting at first, but on second glance you might notice that there are a lot of constants in it. If you substitute the constants with their respective values you will end up with a much simpler formula.

motorSpeed = xSpeed * aConstantValue - ySpeed * anotherConstantValue + angularSpeed * yetAnotherConstantValue

This formula is not only much simpler, it is also very efficient to calculate, just three multiplications and two additions. A NXT can do this in no time. But remember, these constants are not the same for all the motors because each of the wheels has a different wheelAngle. But, you could also have wheels of different sizes, or differences in any of the other aspects. This means that you will have a formula for each of the motors, each formula is the same in structure but has its own constants. These constants can be stored in a matrix where each row in the matrix contains the 3 constants belonging to a single wheel. The matrix has a row for each of the wheels. If you then take the speed vector and multiply this with the matrix then all formulas are calculated at once and the result, the motorSpeed, is stored in a new vector. Each row in this vector holds the speed of a single motor. In java this matrix multiplication would look like this:

Matrix motorSpeed = kinematicModel.times(robotSpeed);

Wow, now things look simple at last! This is the beauty of matrix algebra.

The same kinematic model can be used to transform robot acceleration into motor acceleration. I use this to make my robot accelerate very smoothly. (the regulated motor class of Lejos supports regulated acceleration).

## From tacho counter to robot position

To drive a robot this kinematic model works perfect. But I also want to be able to do things the other way around. I want to be able to calculate robot position from encoder values. At first I couldn’t figure this out at all. The math was just too complex for my mind. That is, until I realized that I just needed to use the inverse of the kinematic model.

deltaRobotPose = deltaMotorPosition * inverseKinematicModel

Here deltaMotorPosition is a vector containing the change in encoder value of each of the motors since the previous measurement. The inverseKinematicModel is the kinematic model inverted. And deltaRobotPose is the change in pose (x and y position and heading) of the robot. Looks simple, doesn’t it? The problem is how to calculate the inverse matrix of the kinematic model. I can’t tell you, because I don’t know. But hey, somebody else already programmed this in Java. I just used the inverse method of the Matrix class.

## From the robots coordinates to the worlds coordinates

There is just one more thing to it. The robot can have any heading, this means that x and y coordinates of the robot are not aligned with the x and y coordinates of the world. To be able to steer the robot to a certain position in a room one must be able to convert this location to a location as the robot sees it. The same goes for keeping track of pose. We have seen the formula to calculate change in pose from the wheel encoders. This change however is a change as the robot sees it, not a change in the robots position it the world. The translation from world coordinates to robot coordinates can also be done with a simple matrix multiplication using a rotation matrix. The rotation matrix itself can be calculated from the heading of the robot.

Suppose you want to drive your robot to the left side of the room. The speed matrix in world frame would look like {0, speed, 0}. this can be multiplied with the rotation matrix to get a speed matrix as the robot sees it.

RobotSpeed =worldSpeed * rotationMatrix

If we want to go the other way around, to get the change in pose in world frame we multiply the change in robot frame with the (you guessed it) inverse of the rotation matrix. For rotation matrices the inverse is the same as the transpose of the matrix, the transpose is far more efficient to calculate.

## Wrap up

This really is all there is to driving a robot. To sum it up. You have a kinematic model to translate robot speed into motor speed. You have a rotation matrix to translate things from world coordinates to robot coordinates.

The same goes for odometry. You have the inverse of the kinematic model to translate change in encoder values to change in robot pose expressed in robot coordinates. You have the inverse of the rotation matrix to translate change robot pose in robot coordinates into world coordinates.

The kinematic model is a constant, it has to be calculated only once (unless your robot changes shape). The rotation matrix on the other hand has to be updated every time the heading of he robot changes.

## The implementation

the robot uses lejos as its brains. Lejos has some excellent classes to drive the NXT motors. The regulated motor class that I used is able to rotate a motor at any given speed. This speed is maintained no matter what the conditions are. It also supports setting an acceleration rate. This is very good for my robot, as for most movements the wheel speed of the three motors is different. If all wheels would accelerate equally, then the slower moving motors would reach their target speed sooner than the faster spinning motors. This results in a robot that shivers and shakes during acceleration (or breaking). All this can be avoided by setting an acceleration value for each of the motors. The ratio of the acceleration values must be the same as the ratio between the (difference between current speed and) target speed of each of the motors. If done properly the robot accelerates very smoothly without jerky movements.

Lejos also has a Matrix class that helps to perform matrix algebra. I used this class to store the (inverse) kinematic models and the rotation matrix. I subclassed it to make a Pose Class that can be used together with the matrix class.

To create the kinematic model I developed a Builder class. This class has all kinds of methods to describe the robot, the wheels the motors and the gear train. When you are done describing the robot, the builder class delivers you a kinematic model and an inverse kinematic model.

To drive the robot I made a kind of pilot class. I plan to discuss it in a future post. This pilot accepts the kinematic model in its constructor.

For odometry I made another class, the Odometer. This class uses the inverse kinematic model.

##